Resilient shell



Patented Sept. 12, 1950 This invention relates `Aizo/resilient slcc-11s,?andv morefparticularlyfto, resilientl metallic shells suit- V liable for compressingiorv .pumping elements, --motor vessels, vibrating members, for the storage lof 9 (curar-rst)- jiparableproblemsxare'"encountered inthe Iluse Eof i'Iuhave :found that thinewalled shellsfeshaped iasfsuriaces ofnrevolution, 'produced by rotation ar'o'und theaxifs 'of symmetry are :excellent ffor the above stated uses without being subjctf-to 'Y the-"above statetl'"diiculties. vBecause"bf'thor- :"ough'symmetryfandi gradual"bhangesincurvature vpresent -inventionwillbe described` morepartic-lo fandbecause'the loading-'involved isfsymmetrical `larly Withreference.,tovftheir use" as pumping- ,elements y K l y Heretofore ilexible metallic members havebeen .used -forrpumping .purposes but fatigue failures to the axis, any unbalanced strains and localz'excessivebending stresses 'are E'eicien-tly avoided. The fexclusio fof any "permanent fcleformations "and". the 1Vprcncer i'consideratiorif cfvwfatigue #limits I in the metals 'of Vthe flexible membersvhaverpre-.zfq's ,-'h'aver'found"'further reduire flthatrthe'zcyclically lsventedserious Vproblems in the practical. application of vthe ieiible members. vTheflexible vmem- ,bers heretoforeused have usually taken the shape 'of "flat discs, conical 'diaplfiragmsv or vorveXp-ansi- :recurrent finechanic'alf=.deformations *of the fthin- `-Walledresilient shell'forrcliamber 'piston should -v'be 'of `a 'substantiallyfinextensive` nature. vThis requiresiproper i Tfree meridional length in :relae, ble .and .contractible Ycorrugateder'neta'llic' Jvesselsgo tion to the size of theflinearst'rcke:employedand commonly known as' bellows, theiiatfdiscsand c.: conica-l'` diaphragms 'being smooth lor .corrugated coneentricallywith theiria'xesvof symmetry.v `In the case of flat rnetallicdislcsan'dconicalvmetalpr'visionforsuitable longitudinal resilient'eXtension. CLInhavei-'found `that `smooth 'walled-.shells 4'ic'zirmotfiineiet these requirements unlessfof-exces sive size and to overcome thisfdircultyv'have ,licv .dianhmgms when-used fon mlrripnsl purposes 25 discovered fan effective structure 1in which" the "their meridional and longitudinal stresses reach the permissible. fatigue limits of 'the metal lat .ip'rlcto'ally insignificant 'd'ctihs' and pressure loads even when. made ,f'the' 'most suitable `metff Wallssorthe fshells are'ccorrugated with a? plurality foffequidistajntly: "disposed meridionali corrugati'ons :'havin'gf-their-maximu-m dimensions at'- th'e-4 equator fand"tapering down -toWard-eacht pole-tof thefshell.

'alsand alloys. "The bellowsfprovide greater lifeigO-v-fsuch-.agconstruction-fagthiSf-zpmvideS-ample,1011- and longer strokesbut Vdo notobviatel'the Yproble`m 'of fatigue failureparticularly -Whererelai'iv'ely,"rapid operation Vand relatively large dis- `placements are? desired.

,Disos and. diaphragms,-when used for.jgzurnping.35

are subjected to,considerable"tangential thrusts vin `,concentric ,regionsY depending on` the imagni- "tude of 'the' central 'deflections which cause in -lthese longitudinal zones of ,.'instab'ilityia more A'git'udinal resilient extensionf-vvhereineeded1toffoiffset excessive longitudinal stresses provides *van Vfincreasedisturdinessand liminatesanycomprescsional1strains or thrusts `in .-fav longitudinaledirection; thusa-precludingfany equatorial buckling.l

fAmon'g the surfacesrcfE revolution z I. have fou-nd fthe?oblate-.elasticoidr' theoblate l'ellipsoi'd with :an iaxesratio v:cfu-A1112; the prolate Lellipsoidi-:and :the .lltoroidsuitable for -theabove described @usesand 0'1' Aless pronounced tendency t0 local` Wavinaim 'having inherent properties to best resist @the 3 Ibuckling orvfolding. This 4unstable buckling constantly changing it's-,place inthe courseof the vibratory deflections is one-of thefmain Acauses fof the fatigue ofthe metallic materials employed above described stresses.

. `1f-It istaecording-lyfan object' of my invention-@to prvrvicle;i ai novelfresilienteshell suitableffor pumpffinggpurposes or.- for i the-storagelof gases, liquids,

and :of theiL-'localbonddisintegration. .Enden-.45fforsolids. l v

its-.persistent factioniinescra`c zksfare startedfeither on' the surface Wherevminutene notchesfand sharp-,edged .grooves are. always present, or within the bulk of the. material itself where minute fsAnothen-obgect-L of my invention Y is to provide flaffnovel-reslient vshell 2- i-n which .fatigue failures fof .themetalliematerials` employed -areovercorne v, .Ariother4 obectfoflmyl-invention is toiprovlde crystallite or intro-crystallite imperfections are` T5fa'-novelresilientfshell-in Which-the meridional the aggravation of the local stress-concentrations Matthesharp activeaedgesaof the wideningfcra'cks as a contributingffactor, atprogressive failure or @and i longitudinal stresses do .-not-f.- approach :the -fatiguefli-mitof the metallic -materialsemployed i the range fof 4deflections; and A.pressure loads ,ei-isualily encountered.V

del-ayedf, fatigue i ruptures-is quite probables :Coms 55;- Anothen object or=my invention.` is to? .provide a novel resilient shell in which tangential thrusts are reduced to preclude local wavy buckling.

Another object of this invention is to provide a novel resilient shell in which local bond-disintegration of the metallic material is precluded. Another object of this invention is to provide a novel resilient shell shaped as a surface of revolution produced by rotation of plane curves of proper concave contours rotated vabout the axis of symmetry capable of resisting unbalanced strains and local excessive bending stresses.

Another object of this invention is to providej a novel resilient shell shaped as a surface of revsion at the equator to preclude any equatorial buckling.

Another object of my invention is to provide a novel resilient shell which is-cheap to manufacture, durable and certain and positive in 0pera-,.

tion.

Other and further obiects of mv invention will appear as the description thereof proceeds.

In the attached drawing, one embodiment of mv novel resilient shell is shown for the purpose of illustration only, and should not be construed as a limitation of the present invention, reference being had to the appended claims to deter- Y mine tbe scopeV of the present invention.

In the attached drawing, Figure 1 is a too view of an oblate elasticoid showing the meridional corrugations; and

Figure 2 is a perspective View of the oblate elasticoid of Figure 1.

In these figures, in which like reference characters indicate similar parts, there-is shown an 'oblate elasticoid. An'oblate elasticoid is a surface of revolution traced by a particular Eulers elastica in which the sides of each curve branch are perpendicular to the axis of symmetry, the curve having points of inexion at these intersections.

The ratio of the axes of the oblate elasticoid is pump stroke, cannot be utilized because for a given length of the meridional perimeter the shape of the oblate elasticoid of revolution has the maximum volume among all other possible shapes of solids of revolution, which, at the comparatively small strokes required at a substantially inextensible deformation, results in a smaller volume at Y the end of the stroke than at the beginning of the same. However, due to the same geometrical property of the maximum volume a greater positive volume displacement is had by using the oblate elasticoid as the shape of the resilient shell at the beginning of the stroke instead of at the end of the stroke. In these iigures I0 represents the surface of a suitable oblate elasticoid having an axis of symmetry II-I I. The surface I0 of the elasticoid is suitably corrugated in a plurality of equidistant meridional corrugations I2 having their largest dimension at the equator I3 of the oblate elasticoid and tapering down toward each pole I4 of the oblate elasticoid to their smallest dimension as at I5, a band of uncorrugated metal I6 being left symmetrically about each pole I4 for f clamping purposes and cut away circularly about poles I4 and I'I to receive the actuating members not shown.

As an example of suitable relative dimensions and characteristics for a resilient shell made of C r-Si-Steel having the shape of an oblate elasticoid of revolution and suitably corrugated as above described, the following dimensions and characteristics are given:

Minor axis-7.353482 cm.

Major axis-12.2748 cm.

Wall thickness-0.01524 cm.=0.006"

Diameter of the cut away portion 17-6.039 cm.

circumference of the clamped portion 11i- 18.9735

Axial distance between the clamped portions 16 of the shell at the beginning of the stroke--4 6.85982 cm. Over-all (equatorial) diameter of the shell at the beginning of the stroke-12.2748 cm.=4.832" Circumierence of the shell at the equator at the beginning of the stroke- 38.5624 cm.

Free meridional length of the shell-2 10.017 cm.

Number of meridional corrugations- 100;

Depth of the meridional corrugations-0.1 cm.

constant;

Meximum width of each corrugatlon at the equator-0.3856 cm.

Width of each corrugation at the boundary of the clamped portion 1li-0.1897 cm.

Maximum cross-sectional moment of inertia of a single meridional strip- 8.37 10-6 cm.

y Minimum cross-sectional moment of inertia of a single meridional strip-4.97X 10-s cm. Assumed stroke-0.4 om. Thrust required for the stroke of 0.4 om. (empty shell)-14.592 kgr.

t Assumed thrust-20 kgr. (for the empty shell) ;v

Power required to run the empty shell at 500 strokes of 0.4 cm. per minute- 0.00444 metric H. P. or 0.00438 H. P.

Assumed power required to run the empty shell at 500 strokes of 0.4 cm. per minute-Vino H. P.;

Horizontal equatorial deflection of the shell at the end of the stroke-2 0.1365 cm.

Over-al1 (equatorial) diameter of the shell at the end of the stroke-12.5478 cm.=4.94 Circumference of the shell at the equator at the end of the stroke-39.42 cm.

Increase of the equatorial circumference at the end of the stroke- 0.8576 cm.

Volume of the clamped (cylindrical) portion of the shell at the beginning of the strokevcc=196-52 cub. om.

Volume of the ring portion of the shell at the beginning of the strokevrc=429.8 cub. cm.

Total volume at the beginning of the strokevtc=vcc+vfc=626-3 cub. cm.

vVolume of the clamped (cylindrical) portion of the shell at the end of the stroke-vce=185.06 cub. cm.

Volume of the ring portion of the shell at the end of the stroke-12re=436.7 cub. cm. Total volume at the end of the stroke- .museum Locationfof the maximum i tensile 1 stressat'ithe clamped boundaryfoftheishell; A Direction 'of this, maximum'.stress-meridionalj lMaximumf tensile4 stress`fkmnax=350 15X internal pressure in kgr./sq. cm.; y v, Them permissible alternating vbending stressfor heat-treated Cr-"Si-Steel (0.8% Cr, 21.5% I Si, `(A).fl 5,%l) is where `kzit-denotes the ultimate alternating cbending :stress of a gpolishe'd' f specimen, V4:Sc-stile `basic safety factor `and iN-the :'notchasafety iactor (notches,scratchesaetc....beingfsomeiofsthe :main technologicaljzhazards). iThe Copper-2% .zBJ alloyi `hasapproxi-mately ,the ;same;.permissible alternating bending'stress. Henceptheishelldescribed above can .safelystandeatltheend.ofthe -sstroke an .internalipressure off-.aboutfal -kgn/sq.

cm.=56.77 lbs/sq. in.

.The :delivery rate of'the i shellv maya-beincreased by a slight increase inV thestroke or byfincreasing the number ofstrokes per fminutebr byzboth.-

;The-metals fand-alloys: suitable ;for the expansible and contractible resilient chamber piston zshells: just described. should-be softiand: comparatively pliable in the annealed .-:state @in-border to ffacilitate the production of @the fshapes `mv'thoiiit any appreciable inherent#anlorzresidualastresses aand; yyetzbe ca .loablegloy suitableheatetreatmentaf f beingnconverted into; highly elastic; resilientgfhard materials of high tensilefstreng'th fand .fatigue limits withl-an elasticlimitof atleast-5000 kgn/nsq. cm. Some of the alloy-steels `forisprings: such as Cr-Si-Steel, Cr-Ni-Steel, Mn-,Si-Steel, Cr-Ni- Mo-Steel, vete. v.are suitable a las. materials .for the above described resilient shells. Due to corrosion these metals should be l.givena protective surface coating of Bakelite,..".g.1yptal, etc., orlin lcases Where corrosion-may 'be present Stainless .steel alloys or non-ferrousalloys shouldbe use'dsuch 'as K'Monel with '315% or vthe coppertberillium alloys with `2'-2\.5 "B., ."CuNil-Beetc. whichafter `proper 'heat treatment, are exceptionally durable, resilientand tough With'high -fatigueliniits.

I'have further found that Whenfthepresilient shells of thepresent'invention are in usehard ,clamping of the shell endsintrodus 'additional r "fforces andv bending` movementsespecially ,when the shell is subjected during deformation-to'internalgpressures. Although it is probable .that these 'supplemental' stresses are confined luto a narrowzone adjacent theboundary theymay in icasescofhard. `clamping aloe.l :materially i. increased and produce a pronounced. distortion of the shell during operation. 'I have Vfound that these Istresses may be minimized by connecting the shell tothe rigid,drivingcparts-with.acushioned clamp- 'suitabieionnseeasz my novel resilientshellsgand @among these fwas :mentioned the torcid. The @toroidalgshapeihassomefstructuralfadvantagesn :respectsto-thezfreemeridional lengthfand-clampwing arrangements. As .the .-differentialfaequators :describingi thelelasticgconditions of; a strained .aan- :nularcsliellzzperxnit .an :exact solution, 1 the f. deflecl.tionsand-stresses f: the .toroid i shell4 can; thus be determined with a high degree oftaccuracy.

ltr-.istoll-be tiezqiresslyA understood. that .thefrefsilient meta'llidshell ofgthe present inventionmay Avbecofeariysuitable sizesfand' the illustration A'here- 'zinbeforesgiveneis for the purposes of illustration only. Thus the inventiongmayfbe .embodiedin containers o .a wide variety of sizes when employedcasreceptacles;for; gases,. liquids, or '-solids. Also, when' used' asa-pump or'compressor, the size may beselectedl'toconform with the `capacity ldesired. ,Itisealso tof. be understoodthat ,the .cor- .rugational characteristics of the shells l.mayube ivarie'dioesuitthe .varyingconditionsof use. It WilL now L .be 4.apparent that. the ...presenta in- ,ventijon provides .a.novel .resilient .shell .suitable {for- .pumping, Nstorage of gases, liquids, or .solids andothergpurposesrshapedasf asurface of. revolu- .-tion.. of .a, planecurve of proper concave: .contour rotated about its .axis of symmetry fin which y.fatigue failuresoithemetal of which it-isffor-med are reduced, inwhich the.` meridional .and longi- '.tudina1 .stresses Ldo 4.not .approach -the `fatigue limiteonthemetal in the .range of; `deiiectionand pressure L.loads .usually encountered, in .which tangential .stresses are. re duced .to preclude. local wavy bucklng,.in. `vvhichv localbondedisintegration ,isprecluded,.whichisnapable of..resisting unbalanced -strains .and A.local .excessive .bending lsti-.esses,and .which 4ismeri'diorially corrugatedsto ,precludeequatorialbuckling, at .the ISametime .-b'e, `ing durab1e,l. cheap .to.manufacture, 4an'decertain and,positive'inroperaition. 1 In. order that mylinventionmay. bemoreclear- .ly,understoodbyathoseskilled in ,the art,'..I. have .madeva .mathematicalanalysis .of .the forcesovinyvolved ,intl-.heY operation., of an oblate .elasticoid The mathematical .treatment is reproduce'dbevloW,.'from Whi'chit .will be .seen that thereonclusions Stated above are justified. I have chosen the oblate elasticoidfto. illustrate the methemati- Acal -treatment because of the importance of this y"f{ )rx'nfunyl invention.

The introduction oftlie above ementioned meridional corrugations, 'considerablyreducing the '.longilnidinal'rigidity, results `in an important sirnv*pliflcation of `the jelastic shell problem :as such, `since', with "the longitudinal V'stresses safely fdisposed of, the influence of anytwo adjacentfjme- 'ridio'nal "strips Yof 'the' shell onseach other, at'the axially .symmetrical loadings "here j involved, is 'greatlysuppressed tlt is," therefore, 1 permissible, Yas a/rsvtiapproximation, Lto regard' such meridiionallynorrugated shells as an agglomeration of a number of thin bars of'variable cross-section, f curved "correspondingly to the meridional shape of therrspec'tivee'shell, thus-"avoiding the` extensive,'1 involvedfand eoemplicate'd# evaluating 'determinations "'of hypergeometri-c',*Bessels 'or Fuchs 'functions'. v`Comparatively fair approximations of nlf'theihorizontal de'ections of= the shell duet rthethrust 'during 'the/compression 4strokef asiwellsastthelextent of the effort 'needed to effect the stroke-deformation Aof ithe empty shell, can 'thus be determined with: a "degree yof accunacycs'ui'ciently: correctfor :thepractical-purposesr byutilizing;therrelatively.I simple7 strainr len*- ergy expression applied to the equationrofcthe chosen meridional curve of the respective shell with the proper consideration of the variable moment of inertia of the individual curved bar cross-section and neglecting any shearing. The determination of the radii of curvature at any chosen point of the given deformed shell results from the fact, that in the case of an inextentional deformation, the measure of the curvature remains constant.

A further simplifying assumption-that of the linear cross-sectional stress distribution, is here also justied, due to the invariably large ratio of the radius of curvature to the wall-thickness of the shell (80 or more).

EVOLUTION OF EQUATIONS PERTAININ G TO THE OBLATE ELASTICOID L. Euler studied in his De curvis elasticis, methodus inveniendi (1744) the buckled shapes assumed by an elastic slender rod with free ends subjected to compression along its unbent--stationary-geometrical axis by two equal oppositing forces, each applied at a corresponding end. He called them the elastic curves and recognized nine different periodical shapes of same, symmetrical in respect to the stationary geometrical axis, depending on the angle quo-under which the individual curve crosses said axis. Because of our particular interest in an oblate lform We will consider only that shape of the Eulers elastic curve for which the sides of each curve-branch are perpendicular to the stationary axis, that is, for which the angle po=90, the curve having points of inflexion at these intersections. l I The derivation of the formal equations of the elastic curve has been exhaustively treated by W. Hess (Ueber das Gyroskop, Mathematische Annalen, vol. 19, 1882, pp, 121-154', and Ueber die Biegung und Drillung eines unendlich dunnen elastischen Stabes mit zwei gleichen Widerstanden, auf dessen freies Endeeine Kraft und ein um die Hauptachse ungleichen Widerstandes drehendes Kraftepaar einwirkt, Mathemasche Annalen, vol. 25, 1885, pp. 1-38). We will, there'- fore, confine ourselves to the interpretation and application of same to our particular problem.

N citations y-the semi-independent variable, the Y-axis of the Cartesian coordinates being the stationary geometrical axis mentioned above,

.r--the dependent varia-ble of the Cartesian coordinates the point of origin of which lies at first in the pointl of inflexion of the elastic curve;

.s-arc-length of the elastic curve, considered as the independent variable; v

S-arc-length of the branch of the elastic curve between two successive points of inflexion, being the points of intersection of. the. elastic curve with the Y-axis;

. p-the angle between the tangent to any chosen point of the elastic curve and the Y-axis;

qntthe angle between the tangent to the elastic curve at its point of intersection with the Y- axis, being also the tangent at the point of inflexion of the curve; in our case pn=90 xmas-maximum value of the abscissa as;

yx=mexva1ue of the ordinate y corresponding to the maximum value of :11;

w-the argument of the elliptic functions;

m-a constant for the particular elastic curve;

c-curvature at any chosen point of the'elastic curve;

p-radlus of curvature at any chosen point of the elastic curve;

Rm--radius of the meridional curvature at any chosen point of the oblate elasticoid of revolution, traced by a branch of the elastic curve rotated around the Y-axis;

Ril-radius of the latitudinal curvature at the same point as Rm of the oblate elasticoid of revolution;

Velasvolume of the oblate elasticoid of revolution, traced .by a branch of the elastic curve rotated around the Y-axis;

Venvolume of an oblate ellipsoid of revolution;

Amar-surface area of the oblate elasticoid of revolution;

Aem-surface area of an oblate ellipsoid of revolution;

Sin am--the Jacobinian doubly-periodic elliptic function Sine amplitude;

Cos am-the Jacobinian doubly-'periodic elliptic function KCosine amplitude dn am-the Jacobinian doubly-periodic elliptic v function delta amplitude;

.2n-the J acobinian zeta function;

K-the complete elliptic integral of the rst kind (Legendre);

K'-the complementary complete elliptic integral of the rst kind (Legendre) Ker-the complete elliptic integral of the first kind for po/2=l5I which has a numerical value K45=1.8540747=K45;

E-the complete elliptic integral of the second kind (Legendre);

E-the complementary complete elliptic integral of the second kind (Legendre) E45the complete elliptic integral of the second kind for po/2=45, which has a numerical value E45=1.3506439=E45;

k-the modulus of the elliptic functions, which for gpo= has a value Seu-perimeter of the ellipse;

yoa--semi-major axis of the ellipse;

b-semi-minor axis of the ellipse;

e-the numerical eccentricity of the ellipse;

lf-the angle between the semi-minor axis of the ellipse and the focal radius drawn to the end of same;

to, t1, t2, t3, and ti-the ordinate spacing values fo the numerical evaluation of denite integrals according to the method of Gauss;

Ao, A1, A2, A3, and Ali-the values of constants for the numerical evaluation of definite integrals according to the method of Gauss.

yin-fthe variable ordinate of the shifted Cartesian Asystem of coordinates, the point of origin of which is moved to correspond to =max.

The formal equations of the elastic curve are:

Squaring the Equation 5 we have:

(i-Cos arcos? am(u+K)=i (e) -Itisiinsgeneral'z u cps @dos (7) l gos Equation 2 may also be presented as followst I 5t or de: "d'cos-vw and the complementary modulus i c-Ell 25we have: l Y

k -Cos 25u,

which, put in (9). willfurnish:A

effluenti-zoem@merkel 17) da: vlL *1de The relation between the, 5Fl-' r 1 395i functions givesjgusg. r y

k2.cos2 am u+K -l-lwdnmmw-l-K) l, Therefore (Nittakesitmhapea,

t sse di J acobnian elliptic After intgratngilotissideefftheEquation 19 We obtain:

aw n A Y, 2'Sm #24440? f (12) FMIefeadMWtmKu-dw Substituting(Ilo'ina) weihavee;l f 5D (20 )f Tne-meerofzftnemanubixiiameuipnefarmeniens furnishes-1use-withi:` l

Where @am denotes the incomplete elliptic K gral of the second ki and, because of Considering (23) and (24), the .Equation 22 will be now:

Putting in the numerical values of the complete integrals we have:

Because the Jacobinian doubly-periodic function Cos am(u+K) has a maximum value for u=K the dependent Variable :c (see the Equation 14) has also for u==K a maximum:

Since the value of the J acobinian'zeta-function Because of symmetry of the elastic curve in respect to these coordinates max. and yx=m4x., they can be regarded as semi-axes of same, especially when the origin of the Cartesian coordinates is moved to the point corresponding xmax.. Such transfer changes the Expression 25 as follows:

y={o.8472131 -0.4569466u'2zn(u+K)} (30) Ratio of the semi-axes of our elastic curve is:

z-max We can express now the arc-length S of our elastic line in terms ofthe semi-axes utilizing the Legendres relation between the complete elliptic integrals of the rst and second kind:

12 For 1c=Sin 45==1c, .as it is in ourcase. there are the following equalities:

, hence,

K-(z-E-K) J2- Considering (33) the Equation 28 will be:

The half-period of the elastic curve u=2K. We have from Equation 4:

being l1 or, with reference to (34):

Utilizing (3l) the Equation 36 may be changedinto:

(37) or, again,

zxnux S4..31'688-6254 S=2.622059xmu (38) Let us consider now theoblate elasticoid of revolution traced by our particular elastic curve while rotated around its Y-axis, and express the ratio of radiuses of the principal curvatures by the elements of the curve, that is the ratio of thev radius of the latitudinal-to the radius of the meridional curvature at any chosen point of the oblate elasticoid.

An obvious trigonometric relation gives us:

'I! lgis"qnstant -value of the-ratio; 4195 is the. 1.5;,j reason or my selection oir-,the .oblatetelaStcoicialV o she, @Mh-ih shelll Containers subietedl i0 an* vl :Qrsmj'k 1c-Sm 45 internal.- 1` ressure,-because cf-the disappearance sin umu of; latitudnabstresses,inftheir wall" Thus, Cos am(ul-K)=--f'fg` in agontainer, all/ ,surfacepoints of whichgsatisfy ,u tleltQjfl) alypai Ofj'djcent.Clmenta?? Therefore, the Equation 49 can be transformed nier; nal stripsoffthe shell:` are exertingneither vas follows;

press pull', nor;bendir 1g.:cri; eachgother, but,v as, eoterbalancegto the internal pressure, they u=k 4 Siinf lily bearfmdependesily andequecvely 25:; v=8'`k2f[--1\.Sl112m] .du nier ionallydirectedbendin stresses; 'I ma du amu Th chime offpur obla tg ,el asticoidj` `f-revolu "0 i u-k 6 4 V--S-skj .du (51) We accomplish the numerical evaluation of the f denite integral or, because of; sgnimetry infres'gect tcggcggg,

011, aitgrgperforminggthe indicated diiererrtiation ,-l; .where t is our new independent variable, which xpressioqinlk i5-iis going to receive n=9 different prescribed values ',Lk "Lto, t1, (-t1),t2, (-tz), i3, (-ta), t4, and (-4).

f Because of this substitution the denite inte- W50- +1 t l 60k; I K d *A the relations betweenggacobirlian fune- 21;@(0' t (54) J'acobnian Zeta Argument u S111 am u dn am u Cos am u function n u 0.93 v 0. 76697 0. 84017 0. 64169 0.1463815 0.94 0. 77232 0. 83771 0. 63523 0. 1461349 0. 95 0. 77761 0.83526 0.62875 0. 1458473 0. 96 0. 78283 0. 83282 0. 62224 0. 1455188 0. 97 0. 78797 0. 83039 0. 61571 0. 1451497 0. 98 0. 79305 0. 82797 0. 60915 0. 1447404 0. 99 u. 79806 U. 82556 0. 60258 0. 1442910 1.00 0. 80300 0. 82316 0. 59598 0.1438020 1.01 0. 80787 0.82077 0.' 58936 0. 1432735 1. 02 0.81268 0. 81840 0. 58272 0. 1427060 1.03 0.81741 0. 81604 0. 57605 O. 1420998 1.04 0. 82208 0. 81369 0. 56937 0. 1414551 1.05 0. 82668 0.81136 0. 56268 0. 1407723 1.06 0.83121 0. 80904 0. 55596 0, 1400518 1.07 83567 0.80674 0.54923 0.1392939 1.08 0. 84007 0.80445 0. 54248 0. 1384989 1. 09 0. 84440 0. 80218 0. 53571 0. 1376673 1. 10 0. 84867 0. 79993 0. 52893 0. 1367994 1. 11 0.85286 0.79769 0. 52213 0. 1358957 1.12 0. 85700 0. 79547 0. 51532 0. 1349563 1. 13 0.86106 0. 79328 0. 50850 0. 1339819 1.14 0. 86506 0. 79110 O. 50166 0. 1329727 1. 15 0. 86900 0. 78894 0.49481 0. 1319292 1. 16 0.87287 0. 78680 0.48795 0. 1308518 1.17 0. 87668 0. 78468 0.48108 0. 1297409 1.18 0. 88042 0. 78258 0. 47419 0. 1285968 1. 19 0. 88410 0. 78050 0.46730 0. 1274201 1. 20 0. 88772 0. 77845 0. 46039 0. 1262112 1. 21 0. 89127 0. 77642 0.45348 0. 1249704 1. 22 0. 89476 0. 77441 0. 44655 0. 1236983 u1=, 1. 227632 0. 897370 0. 772881 1.23 0; 89818 0.77242 0.43962 0. 1223952 1. 24 0.90155 0. 77046 0.43268 0.1210616 1. 25 0. 90485 0. 76852 0. 42573 0. 1196980 1. 26 0.90809 0.76661 0.41877 0.1183047 1. 27 0.91127 0. 76472 0.41180 0.1168823 1. 28 0. 91439 0. 76285 0.40483 0. 1154313 1. 29 0. 91745 0.76101 0.39785 0.1139519 1.30 0.92045 0. 75920 0.39087 0. 1124448 1. 31 0.92338 0. 75742 0.38388 0. 1109104 1- 32 0.92626 0. 75566 O. 37688 0. 1093491 1. 33 0. 92008 O. 75393 0. 36988 0. 1077615 1. 34 0.93184 0.75222 0. 36287 0.1061479 1. 35 0.93454 0. 75055 0.35586 0. 1045089 1. 36 0. 93718 0. 74890 0.34884 0- 1028450 1. 37 0.93976 0.74728 0.34182 0.1011565 1. 38 0. 94229 0. 74568 0.33480 0.0994441 1. 39 0.94476 0. 74412 0.32777 0.0977081 1. 40 0. 94717 0.74259 0.32074 0. 0959491 1. 4l 0.94952 0. 74108 0.31370 0.0941676 1.42 .0. 95182 0.73961 0.30666 0.0923639 1. 43 0. 95406 0. 73816 0. 29962 0.0905387 1. 44 0.95624 0. 73675 0. 29258 0. 0886924 1. 45 0. 95837 0.73537 0. 28553 0. 0868254 1. 46 0. 96044 0. 73401 0. 27848 0.0849384 1. 47 0A 96246 0. 73269 0. 27143 0.0330317 1. 48 0. 96442 0. 73140 0. 26433 0.0811058 1. 49 0. 96632 0. 73014 0.25733 0.0791613 uz=1. 495655 0. 967372 0.729451 1.50 0. 96818 0.72892 0.25027 0. 0771987 1. 51 0.96997 0. 72772 0.24321 0.0752184 1. 52 0. 97172 0.72656 0. 23615 0.0732209 1. 53 0. 97340 0. 72543 0.22909 0.0712068 1. 54 0. 97504 0. 72433 0.22203 0. 0691764 1. 55 0. 97662 0. 72326 0. 21497 `0. 0671304 1. 56 0.97815 0. 72223 0.20790 0.0650693 1. 57 0. 97962 0. 72123 0.20084 0.0629934 1. 58 0. 98105 0. 72026 0. 19377 0.0609033 1. 59 0.98242 0. 71933 0. 18671 0. 0587996 1.60 0. 98373 0.71843 0.17964 0.0566826 1.61 0.98500 0. 71756 0. 17257 f 0.0545530 1. 62 0. 98621 0. 71673 0.16550 0.0524111 1.63 0.98737 0. 71593 0.15843 0.0502576 l. 64. 0.98848 0.71516 0.15137 0.0480928 1.65 0.98953 0. 71443 0. 14430 0. 0459174 1. 66 0.99054 0.71373 0.13723 0.0437318 l. 67 0. 99149 0. 71307 0. 13016 0. 0415364 1. 68 0.99240 0. 71244 0. 12309 0.0393319 1. 69 0.99325 0. 71185 0.11602 0.0371186 1. 70 0. 99405 0.71129 0.10895 10. 0348972 u3=1. 702069 0.994205 0. 711183 1. 7l 0.99480 0. 71077 0. 10188 0.0326680 1. 72 0. 99550 0. 71028 0.09480 0. 0304316 1. 73 0.99614 O. 70982 0.087773 0. 02818.86 1. 74 0. 99674 0. 70940 0.08066 0. 0259393 1. 75 0. 99729 0. 70902 0.07359 0. 0236843 1. 76 0. 99778 0. 70867 0.06652 0.0214242 1.77 0.99823 0.70836 0.05945 l 0.0191593 1. 78 0.99863 0. 70808 0. 05238 0.0168902 1. 79 0. 99897 O. 70783 0.04531 0. 0146174 1. 80 0. 99927 O. 70762 0. 03824 0. 0123414 1. 8l 0.99951 0. 70745 0.03117 0.0100628 1. 82 0. 99971 0. 70731 0.02409 0. 0077819 u4=1. 824557 0. 999778 0.707265 flo-@(to) :0.3802390686255 :0.226628 Sin am(1.227632) 4 A1150.) JF14-11)]:0.9125417(1.917935 +0.151640) 0.312347.1.96s975=0.615003 Sin am(1.702069)]4 [0.994205 dnam(1.702069) 0.711183 A41@ (i4) -l-( ml:0.081274-(3.992871-1-0000001) 0.081271193992872:0.3241517. o Inserting the above computed Values into (56) 5 We obtain:

+1 ft) dt=2.669034 -1 (in am(1.s24557) 1 dn am(0.029517) and, thus the denite integral (54) du am u The estimated error is less than 0.000005. We can therefor, put:

Insertion of this computed numerical value (58) of the definite integral in (51) results in:

8-7r-k ma Considering (13), (27) and (28), We obtain:

or, after insertion of the numerical Values of the constants:

0.25 V (7rmrxnx`2yz-'m nx) V= (vr-xuZg/MQ0295085247429?, V:(7r`x?nax'2-yz=mux)'0.730127 We may note here that the expression in the brackets represents the Volume of a straight cylinder having a circular base with a radius .rmxthe major Isemi-axis, and a height equal to 2.yx=maY.-tl1e minor axis of the oblate elasticoid of revolution.

We have further:

T.:(xax'I/at-max)'2"l'0.730127` V= mx-ypmx-4-587523 (62) The surface of the oblate elasticoid of revolution is:

u=2k A=21rfw-ds (6s) u=0 or, because 0f symmetry in regard to its major axis:

u=k A=41rfzds With reference to (13) and (4) We have:

` dn am u 8nk2 Sin am 11.

m2 dn am u For lthe numerical evaluation of the definite integral in the Expression J Sin am u A= du 21 we apply again the integration method of Gauss using n-9 ordinates and the values already obtained.

Sin am(0.152005) dn am(0.152005) Sin am(1.824557) dn am( 1.824557) The estimatedl error being less than 0.000002, We can neglect it, and thus put lo J=1.570779 (70) Inserting (70) into (65) we arrive at:

A=4`11k'*(2k/m)1/m1.570779 and considering (13), (27) and (28) we obtain: 4- -k l Azmfyaw--g-lL-1-570779 A"m`y"'m* 0.647213 '1570779 A=zm49am-10.4796661.570779 A=wmxy2mx16461566 (72) For comparison We determine the surface area of an oblate ellipsoid of revolution having same axes ratio as our oblate elasticoid, namely a/b="1.669254 (73) Let f denote here the linear eccentricity:

f=m=5vmr1 74) then the surface area 0f the oblate ellipsoid of revolution is z a-'l-f The volume of the oblate ellipsoid of revolution having similar semi-axes as our oblate elasticoid is:

Ve11.==4/3 .1r.a2.b (78) or, V.u.=a2.b.2.. 0.666667 V.u.=a2.5. 6.263165 .0.666667 7 5 i V.u.=a2.b. 4.166792 79) 2,522,401 A23 24 Comparing ('79) with (62) we see that the volbut, since: Y' ume of our oblate elasticoid of revolution is larger h 5 Cos amw) than the volume of an 'oblat'e ellipsoid of revolu- S111 @mW-FK) =m `((83.) tion lwith similar axes. Furthermore, we may point out here that for a given length of the me- 5 We have: C ridional perimiter the shape of the oblate elasti- Q 4 0S am u cold of revolution has the maximum volume MUM-K) mw) k Sm amw) dn am(u) among all other possible shapes of'solids of rev- S olution, because the equation of its meridional z(u+K)=- zn(u) 192.005 am(u),ma m(u)r, curve-the elastic line-is the one, which fully 10 d amm) satisfies the isoperimetric criteria established by and, considering (80) Euler Sin am(u) For the purpose of practical enumeration of the zn u+ K) =zn(u) -0.,5-Cos a1000-m)- (84) coordinates and y of the elastic curve we may ,n simphfy the Expressions 13 and 26, 15 Insertion of (84) into (30) results in:

1 v Sin am(u) y={0.84721310450940052 21:25@ 0.5005 am(u) dn amm) 1 l iSin am(u)} ym=77l{0.8472131 04509400.77 2v25 u1+1o0s ammdn am@ (85) Utilizing the Relation 50 the Equation 13 The radius of curvature'atany Chosen Point changes to: of the elastic curve is (see Expression 40) m dn amw) 25 The tangent tovany chosen point of the elastic curve forms an 'angle p with the Y-axis. It is x=2.k2%lu) (see Expressions '3 and 50):

" 2 -2 o 2- 80 Cos -05 .S1-DEM (87) h We are givenhelow a tableof the numerical We ave' values of the'Cartesian coordinates of the elas- 1 sin (mw) 35 tic curve-m and ym for m=1, together with the V x= dn (mw) (81) y corresponding Values of the radius of curvature p, Cos p andthe angle p between the tangent and the Y-axis, for theascending values of the The additional theorem of the Jaoobinian zeta independent Variable u.

functlon' 40 Next is shownthe elastic curve itself drawn l if" to al Scale 2.yx=max=8.4:72131" .or ml=lo.l2n, the

k2.Sin am(u .Sin amw) .Sin am(ul-v) (82.) upper Dart of the 'curve :showmg dlvlSlOnS 0f s=0.25 vcorresponding to the ascending steps gives for v=K; y of u=0.05. The circle of curvature, drawn for the apex, is being very .closely followed by the zn(u+K) :211(24) -lc2.S1n amm) .Sm am(u}-K) curve for quite some length.

Argument Grdinate Abscissa Radius o o l u ym z Y p Cos c 7p +90 0.35 0 840009 0 349871 2 858190 0 001205 8029'28" 0.40 0 830543 0 399740` 2 501020 0 079890 8525 3" 0.45 8320 0 449544 2 224454 0 101045 8412' 2l 0. 50 0 820410 0 499220 2 003124 0 124005 8250'31" 0.55 0 819534 o 548741 1.822353 0 150558 8120'27" 0, 00 0 811314 0 598059 1.072070 0 178837 7941'53" 0.05 0 801014 0 047103 1. 545349 0 209371 7754'52" 0.70 0 790337 0 095822 1 437149 0 242084 7559'20ll 0.75 0 777372 0 744104 1 343900 0 270845 7355'41'l 0.80 0 702010 0 791809 1 202835 0 313528 7143'41'l 0.85 0 745990 0 839029 1 191853 0 351984 092329" 0.90 0 727401 0 885439 1 129383 0 392001 0085515'l 0.95 0 700770 0 930979 1 074138 0 433301 0419 8" 1.00 0 084040 0 975509 1 025100 0 475321 0137'11" 1.05 0 059180 1 018882 0 981308 0 519000 584351" 1.10 0 032132 1 000930 0 942509 0 502780 5545' 5" 1.15 0 002888 1.101478 0 907801 0 000027 5239'14" 1.20 0 571409 1.140309 0 870907 0 050221 492033" 1.25 0 537880 1 177393 0 849334 0 093127 40 8'19'l 1.30 0 502182 1 212395 0 824813 0 734951 424149" 1. 3 5 0 404414 1.245140 0 803122 0 775180 391041" 1.40 0 424592 1. 275495 0 784009 0 813444 3533'58" V1. 45 0.383108 1.303249 0 707313 0 849228 3152'20" 1.50 0.349814 1.328239 0.752870 0 882109 28 0' 8H 1.55 0 294901 1 350303 0.740574 0 911059 24915l 'f ,1. 00 0 248711 1 309277 0 731042 0 937459 2022'15" 1.05 0 201281 1 385002 0 721989 0 959198 102525" g 1.70 0 152871 1 397531 0 715547 0 970540 12320l 1" 1.75 0 103099 1 400575 0 710947 0 989220 825' 5" 1.80 0 054029 1 412150 0 708137 0 997092 422'14" 1.85 0.004073 1. 414207. 10. 707110 v y0. 999991 014I o" l l. K-1. 8540747 0.000000 1.414214 0.707107 1.000000 0 0' 0" To those skilled in the art changes to or modificationsof the above described illustrative embodiment ofthe present invention may now be suggested, without departing from the inventive concept of the present invention, and reference should be had to the appended claims to determine the scope of the present invention.

In the foregoing specication and in the claims I refer to a thin walled shell 4and give the thickness of such a shell in a particular example. It is to be understood that by a thin walled shell I mean one in which the stresses in the mid-layer of the wall may be neglected for stress analysis purposes as is well understood and is thoroughly explained in works on stress analysis.

What is claimed is:

1. An expandable and contractible resilient motor vessel comprising a meridionally corrugated thin-walled shell of uniform thickness shaped generally as a surface of revolution of Eulers elastica rotated about its axis of symmetry.

2. An expandable and contractible resilient thin-walled vessel of uniform thickness shaped generally as an oblate elasticoid and meridionally corrugated, said elasticoid being formed by the revolution of Eulers elastica curve rotated about an axis of symmetry with the curve perpendicular to said axis at its intersection therewith.

3. An expandable and contractible resilient thin-walled vessel of uniform thickness shaped generally as an oblate elasticoid formed by the revolution of Eulers elastica curve about an axis of revolution and having equidistant meridional corrugations having their greatest dimension at the equator and tapering toward the poles of the oblate elasticoid.

4. An expandable and contractable resilient thin-walled vessel of uniform thickness shaped substantially as an oblate elasticoid, formed by revolution of Eulers elastica curve about an axis of symmetry, and having meridionally extending corrugations over a portion thereof.

26 5. A vessel as in claim 4 wherein said corrugations extend completely around the equator of said vessel and only partially between the poles thereof.

6. A vessel as in claim 5 wherein said corrugations extend partially between said poles symmetrically about the equator of said vessel.

7. A vessel as in claim 6 wherein said corrugations extend between two equal imaginary circles concentric respectively with said poles.

8. An expandable and contractable resilient thin-walled vessel of uniform thickness shaped substantially as an oblate elasticoid, formed by revolution of Eulers elastica curve about an axis of symmetry, andv having meridianally extending corrugations having a depth small in comparison to the equatorial diameter of said vessel and extending around the equator of said vessel, said corrugations being spaced apart by distances small in comparisonwith the circumference of said vessel on said equator.

9. A vessel as in claim 8 wherein said corrugations are most widely separated at said equator and converge toward the poles of said vessel.

` ALEXANDER RAVA.

REFERENCES CITED The following references are of record in the file of this patent: 

